lecture slides: lecture optimization

1Challenge the future

G-L-5:

OptimisationAdvanced Modelling

Prof. Dr. Christos SpitasDr. Y. Song (Wolf)

Faculty of Industrial Design EngineeringDelft University of Technology


Contents

• Why optimisation?

• A case study

• The gradient descent method

• Discussions

• What did we learn today?


Optimisation@Modelling

Case brief

System

Experiment

Build abstract model

choices

choices choices

choices

do you need more insights/ data?

‘touch’ your thoughts

what are you studying?

what exactly...?

what do you foresee?

what does your model foresee?

did you both agree?everything you did/ saw/ felt/

remember to be true...

Finish!

Revisit choices

Compare

Acceptable?

Experience

Thought simulation Solve/ simulate

Courtesy of centech.com.pl and http://www.clipsahoy.com/webgraphics4/as5814.htm


Why Optimisation?

Learn

Identify, choose Modelling

Solve

Evaluation

System

Product optimisation

In mathematics, Optimisation refers to choosing the element from some set of available alternatives that maximises/minimises a specified metric.

Product Optimisation is not just making a product better in some respect (a criterion / metric), but making it the BEST (optimal) in this respect.


A case study: The coffee cups

Fictional case study, for education only.Douwe Egbert® are resisted trademarks.

Courtesy of http://www.douweegbertscoffeesystems.com/dg/OutOfHome/OurProducts/Coffee/Cafitesse/Machines/

In the new generation of Douwe Egberts® coffee machine, the preheat coffee cup feature is introduced. In the preheating process, water steam from the nozzle preheats the cup to a certain temperature before the coffee is served.

Experiments indicate that: 1. if the cup is preheated to 92°C , the best coffee can be served; 2. 80% of water steam is condensed inside the cup, the rest is absorbed by the air.

Besides, Douwe Egberts® also manufactures its own coffee cups with different sizes and materials, for example, the Hollandsche series and the Standard series.

You are asked to find the initial steam temperature and the amount of steam that should be produced in order to preheat either of the two types of cups as close as possible to 92°C (minimise the deviation).


System identification

Standardcup

Steam

Hollandschecup

Fiction case study, for education only.Douwe Egbert® are resisted trademarks.



Cause-effect

Fiction case study, for education only.Douwe Egbert® are resisted trademarks.


Superheated steamCondensingCondensed


Modelling

min holland holland( 100) (100 ) 80% ( )steam steam stea itial steam steam steam water Hfinal Hfinal initialm c T m L m c T m c T T

Hollandsche cup

Standard cup

min standard standard( 100) (100 ) 80% ( )steam steam stea itial steam steam steam water Sfinal Sfinal initialm c T m L m c T m c T T

Cup is heated

Latent heat

Condensed water cools

downSteam cools

down



Modelling - Choices

min holland holland( 100) (100 ) 80% ( )steam steam stea itial steam steam steam water Hfinal Hfinal initialm c T m L m c T m c T T

Hollandsche cup

Standard cup

min standard standard( 100) (100 ) 80% ( )steam steam stea itial steam steam steam water Sfinal Sfinal initialm c T m L m c T m c T T

We chooseThe density of water is 1000 kg/m3;

The latent heat of water vaporization is 2,260,000 J/kg;

The specific heat of water steam is 2080 J/(kg·K);

The specific heat of water is 4181 J/(kg·K);

The initial temperatures of both cups are 20 °C;

The weight of the Hollandsche cup is 0.20 kg, the specific heat is 750 J/(kg·K);

The weight of the Standard cup is 0.13 kg, the specific heat is 1070 J/(kg·K);

The steam temperature doesn’t change before it reaches the cup;

The complete process happens in a very short time;


Solving/Simulation

Hollandsche cup final temperature

Standard cup final temperature

6steaminitial

steaminitial8320 9.8804 10 15000( , )

16724 750steam steam

Hfinal steamsteam

m T mT m Tm

6steaminitial

steaminitial8320 9.8804 10 13910( , )

16724 695.5steam steam

Sfinal steamsteam

m T mT m Tm

Design parameters steaminitial( , )steamm T


Our wishes

Make the deviation

as small as possible

Make the deviation

as small as possible




You are asked to find the initial temperature and the amount of steam that should be produced in order to preheat either of the two types of cups as close as possible to 92°C (minimize the deviation).

steaminitial( , ) 92Hfinal steamT m T

steaminitial( , ) 92Sfinal steamT m T

Design briefDesign brief

ANDWishesWishes


The metrics

21 steaminitial( ( , ) 92)Hfinal steamM T m T

ExampleThe differences

22 steaminitial( ( , ) 92)Sfinal steamM T m T

( ) 5y x x 2( ) 5y x x


Formulate the objective function based on metrics

2

steaminitial1

( , )steam i ii

f m T W M

Non-dimensional

We want to minimise

2

steaminitial steaminitial1

min ( , ) where ( , )steam steam i ii

f m T f m T W M

The objective function


Formulate the objective function based on metrics

2 2steaminitial Hfinal steaminitial Sfinal steaminitialmin ( , ) 0.5( ( , ) 92) 0.5( ( , ) 92)steam steam steamf m T T m T T m T

0.5iW

In our case study, we choose


Start from a 1D problem: Finding the minimum of a function

( )f x


The gradient descent method

Start with a point (guess)

Repeat•Determine a decent direction•Choose a step•Update

Until stopping criterion is satisfied

( )Guess

f x( )f x






( )Guess

f x

Guess

( )f x

1






( )Guess

Step f x

Guess

Next Step

( )f x

2

1

( )NextSep Guess Guessx x Step f x






1( )

Heref x

Now we are here

( )f x

2






1

2

( )f x

3

45

6

f


2D gradient descent method

x

y

22

),( yxxeyxf ),( yxf



),( yxfx

)),(),,((

),(

yxfy

yxfx

yxf

),( yxfy



Start

Minima





Solving our design problem:The coffee cups

GuessMsteam = 0.003 Tsteaminitial=110

The minimum value of the objective function is 6

It is a compromise

Gradient decent path


Wishes vs Designs




You are asked to find the initial temperature and the amount of steam that should be produced in order to preheat either of the two types of cups as close as possible to 92°C (minimize the deviation).

Design briefDesign brief

steaminitial( , ) 89.5Hfinal steamT m T C

steaminitial( , ) 94.3Sfinal steamT m T C

Hollandsche cup final temperature

Standard cup final temperature

Output of the optimised

design

Put real optimal parameters in side


Generalised form




),...,( 21mn

mm pppf

1 2min ( , ,... )nf p p pTo optimise a function:

),...,( 002

01 nppp

)(),...,(

),...,(

,...,21

112

11

21mn

mm pppmmn

mm

mn

mm

fstepppp

ppp

mstep

112

11 ,..., m

nmm ppp

f


Question! Weights in the objective function

The market share

We can emphasisea wish by increasing

Its relative weight


Question! Weights in the objective function

steaminitial

2 2steaminitial 1 Hfinal steaminitial 2 Sfinal steaminitial

min ( , )where

min ( , ) ( ( , ) 92) ( ( , ) 92)

steam

steam steam steam

f m T

f m T W T m T W T m T

Weight Value = 0.75

Weight Value = 0.25

steaminitial( , ) 88.4Hfinal steamT m T C steaminitial( , ) 93.1Sfinal steamT m T C

Hollandsche cup final temperature Standard cup final temperature

Output of the optimised

design

Put real optimal parameters in side

And compare to previous


Question! Square in the metrics

21 steaminitial( ( , ) 92)Hfinal steamM T m T

Why square insteadof absolute value?

22 steaminitial( ( , ) 92)Sfinal steamM T m T

( ) 5y x x

( ) 5y x x

2( ) 5y x x


Problem: find the maxima

Original function f(x)

With a minus in front –f(x)

Or

Gradient ascent

Function f(x)

Guess


Problems: Guess the starting point

The starting point is not good


Problems: Choice of the sizes of steps

Small steps: inefficient Large steps: potential risks

The starting

point

The starting point


Advanced: Dynamic steps

),...,(

),...,(

21

21

mn

mm

mn

mm

ppp

ppp

f

f

Acquire the direction

Assign the initial step

If the step is too large, reduce the step to

a certain percentage


Problems: The stopping criterion

),...,(),...,( 2111

21

1mn

mmmn

mm pppppp

f

Intuitive criterion

Other criteria

Example 1

Example 2

…

),...,(),...,( 2111

21

1mn

mmmn

mm pppfpppf

2 22 2

1 1 2

N

i i N

f f f ffx x x x


Fundamental problems: Local minima

Local minima

Global minima

x

y )(1.0),(

22)cos()sin( yxeyxf

yx

),( yxf

Top view


Fundamental problems: Local minima

Local minimum

Global minimum

Guess


The world is much larger

Tools forOptimisation

Quasi-Newton / conjugate gradient methods

Box (complex) / Hook-Jeeves

Genetic algorithms (Evolution Strategy)

and many more


What did we learn today?

Formulate the objective function (metric) of your design

An optimum is (almost) always a compromise (competing metrics!)

Gradient descent methodis a possible way to find the minimum

of your objective function

Optimisation methods have limitations


Success!

Prof. Dr. Christos SpitasDr. Y. Song (Wolf)

Faculty of Industrial Design EngineeringDelft University of Technology

lecture slides: lecture optimization

Education

steam c water

steam lsteam

steam cwater

specific heat of water

initial steam temperature

coffee cup feature

coffee cups

best coffee